For a one-dimensional classical Ising model with the Hamiltonian $$H=-J \sum_{i}\sigma_{i} \, \sigma_{i+1}$$ where $\sigma=\left\{+1,-1\right\}$ one can calculate two point correlation for the spins $$\left<\sigma_{i} \, \sigma_{j}\right>.$$ I understand the meaning for this is that how two spins at different positions are correlated or in other words how fluctuations at the ${i}^{\text{th}}$ position affects the the spin at the position $j$.

Now, what is the physical meaning of four point correlation function $$\left<\sigma_{l} \, \sigma_{m} \, \sigma_{n} \, \sigma_{p}\right>.$$ What extra piece of information does it give? Can some explain intuitively?


Let me answer a more general question (which might not be what you are after...): what information is encoded in general correlation functions $\langle\sigma_A\rangle$, where $A$ is a finite set of vertices and $\sigma_A=\prod_{i\in A} \sigma_i$?

It turns out that one can prove (it's actually easy) that, for any local function $f$ (that is, any function depending only on finitely many spins), one can find (explicit) coefficients $(\hat{f}_A)_{A\subset\mathrm{supp}(f)}$ such that $$ f(\sigma) = \sum_{A\subset\mathrm{supp}(f)} \hat{f}_A \sigma_A $$ where $\mathrm{supp}(f)$ is the (finite) set of spins on which $f$ depends.

This means that knowing the correlation functions $\langle\sigma_A\rangle$ for every finite set $A$ allows you to compute the expectation of any local function $f$: $$ \langle f\rangle = \sum_{A\subset\mathrm{supp}(f)} \hat{f}_A \langle\sigma_A\rangle . $$ In this sense, the correlation functions $\langle\sigma_A\rangle$ contain all the information on the Gibbs measure.

(Let me emphasize that everything I said is completely general and not restricted to the one-dimensional model.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.